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Journal of Mathematical Chemistry 27 (2000) 155170 155

Enzyme kinetics of multiple alternative substrates

S. Schnell a and C. Mendoza b

a Centre for Mathematical Biology, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK

E-mail: [email protected] Centro de F sica, Instituto Venezolano de Investigaciones Cient ficas (IVIC), PO Box 21827,

Caracas 1020A, Venezuela

E-mail: [email protected]

Received 28 February 2000

An innovative theoretical approach that enables the complete characterisation of enzyme

substrate and enzymesubstratecompetitor reactions is generalised to systems with multiple

alternative substrates. Based on the quasi-steady-state assumption, time-dependent closed

form solutions are presented for cases with even, weak and mixed substrate competition. The

analytic framework should facilitate the development of computational fitting procedures for

progress curves, simplifying the measuring process and increasing the reliability of reaction

constant estimates.

1. Introduction

The introduction of alternative substrates in enzyme processes is an important

approach in the study of living materials and their transformations, as the inducedperturbations allow a more detailed understanding of the normal initial states. In par-

ticular, alternative substrates are employed to bring out the features of enzyme active

centres due to their selective and competitive reactions [38]. Driven by economic and

environmental issues, much effort has been devoted to the industrial synthesis of enan-

tiomerically pure compounds; in such enterprises, it is essential to estimate the kinetic

parameters of the competing racemases in order to ensure efficient resolution [20].

Classical kinetic methods have been applied to enzyme systems involving no

more than three substrates; e.g., the techniques of Daziel coefficients and Haldane re-

lationships [17], which are essentially qualitative, and the more quantitative approaches

of isotope exchange [4,28] and alternative substrates [710,27,24,41]. Further exten-

sions to multiple alternative substrates are generally avoided since analysis gets rapidly

complicated without adding much information to that obtained by studying the sub-strates individually. An approach to overcome this problem has been proposed [27]

where alternative substrates are taken as competitors of a given substrate thus asserting

that insight into the kinetics can be obtained by evaluating their competitive effects.

Previous work on multiple competition [1,6,15,16,3538] has been mainly con-

cerned with the development of graphical methods to estimate, within the Michaelis

J.C. Baltzer AG, Science Publishers

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156 S. Schnell, C. Mendoza / Enzyme kinetics of multiple alternative substrates

Menten (MM) framework, reaction constants and to elucidate the reactant interactions.

Despite their virtues in data visualisation, diagnostics and education, such graphical

methods can be tedious, of relatively poor accuracy and imply acquired expertise

[6,14,18,25]. The general availability of powerful microcomputers, on the other hand,

should now make approaches based on least-squares curve fitting procedures more

attractive [11,12,39], although the hitherto lack of analytic expressions for progress

curves hampers general acceptance as direct numerical integration can be computa-

tionally intensive [19,22,43].

Schnell and Mendoza [29] have revisited the MM formalism for the case of the

basic enzymesubstrate system, and have found for the first time closed form solutions

that characterise the complete evolution of both the reactant concentrations and their

time derivatives. This analytic framework facilitates the development of least-squares

fitting procedures to determine accurate reaction parameters from progress curves (for

an independent verification, see [22]). Therefore, it proposes a more efficient and

reliable experimental methodology based on the timing of a single decay curve insteadof the usual lengthy estimates of initial velocity as a function of initial substrate

concentration from several decay curves. This theoretical approach has been recently

extended to describe the fully competitive enzymesubstratecompetitor system [30],

obtaining closed form solutions for the three nominal cases of competition: even, weak

and strong [26].

In the present report we generalise the work in [29,30] to the case of multiple

alternative substrates. This is the first stage in a current systematic revision of more

complicated enzyme reaction networks. Exclusive multiple competition is reduced to

an n-substrate reaction system in order to make the mathematical formalism morecompact, and is illustrated with solutions for the case of one and two substrates.

The enzyme kinetics of such systems is briefly reviewed, followed by a derivationof the corresponding analytic solutions for the different cases of competition. Some

conclusions of the work are also discussed.

2. The alternative n-substrate system

The multiple alternative substrates system can be reduced to the scheme

Si + Ekiki

ESik2i E+ Pi (1)

with n substrates (i = 1, . . . , n). E is the free enzyme, and Si, ESi and Pi respectivelydenote the ith substrate, enzymesubstrate complex and product. The parameters ki,ki and k2i are positive kinetic constants for the ith channel. Thus, if in an alternativesubstrate system i = 1 is taken to be the leading substrate, the system contains (n1)competitors (i = 2, . . . , n).

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S. Schnell, C. Mendoza / Enzyme kinetics of multiple alternative substrates 157

Figure 1. Characteristic curves for the enzymesubstrate reaction (1) with n = 1: (a) reactant concentra-tions and (b) time derivatives. The infinite time range has been reduced to the interval (0, 1) by means

of the exponential time scale = 1 1/ ln(t+ exp(1)). The concentrations have been nondimensionallyscaled as follows: free substrate s1 = [S]1/[S0]1; free enzyme e = [E]/[E0]; complex c1 = [ES]1/[E0]and product p1 = [P]1/[S0]1. The time derivatives have been scaled to their absolute maximum value:

s1 = d[S]1/dt; e = d[E]/dt; c1 = d[ES]1/dt and p1 = d[P]1/dt.

For the basic enzymesubstrate reaction (n = 1), a time-dependent closed formsolution for the substrate concentration has been derived [29]:

S

1

(t) = W

S0

1

exp

1t+

S0

1

, (2)

where W is the omega function W (see the appendix), [S

]1 [S]1/K1 is the substratereduced concentration (with initial value [S0]1) and 1 vmax1 /K1 is the first-orderrate constant. The familiar MM constant and maximum velocity are defined, respec-

tively, as

K1 k1 + k2k1

and vmax1 k2[E0]. (3)

Expression (2) describes the substrate decay during its complete duration (0 t < ),and its effectiveness in experimental data fitting is ensured by the availability of highly

efficient algorithms for W [2,3,21]. Moreover, as shown in figure 1, characteristicprogress curves for all reactants and their time derivatives can be henceforth generated.

Similar solutions for the fully competitive enzymesubstratecompetitor system

(i = 1, 2) have also been formulated [30]. Making use of the competition matrixij j/i to classify different levels of competition [26,30], the solution for evenlycompetitive substrates (12 1) is given by

Si=

[S0]i

[S0]1 + [S0]2

W

S0]1 +

S0

2

exp

it+ S01 + S02. (4)

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It may be appreciated that the reduced substrate concentrations keep the constant ratio

[S0]1/[S0]2 throughout the reaction. For weak competition (12 1) the solutions

now take the form S

1(t)=W

S0

1exp

1t+ S01, (5)S

2(t)=

S0

2

[S]1(t)

[S0 ]1

12, (6)

where [S]i [S]i/Ki and i vmaxi /Ki are now referred to as the apparent reducedconcentration and apparent first-order rate constant, respectively, normalised to the

apparent MM constant

Ki Ki

1+[S0]j

Kj . (7)

Therefore, the fast substrate concentration decays in a very similar fashion to the

isolated case, i.e. as apparently alone, but the competition causes an increase of the

MM constant [32]; on the other hand, the competitor decay corresponds to that of the

fast substrate but severely attenuated by a power of 12. Characteristic curves for thissystem are depicted in figures 2 and 3. The last case of strong competition (12 1)is trivial as its solutions are essentially equations (5) and (6) with permuted indices

noting that 21 = 1/12. An interesting result that emerges from the formalism is therole played by the apparent MM constants which is only of noteworthy importance in

systems with uneven competition.

3. Enzyme kinetics

By applying the law of mass action, the time evolution of the n-substrate sys-tem (1) is described by the nonlinear coupled differential equations

d[S]i

dt=ki[E][S]i + ki[ES]i, (8)

d[E]

dt=

ni=1

ki[E][S]i + (ki + k2i)[ES]i, (9)d[ES]i

dt

= ki[E][S]i

(ki + k2i)[ES]i, (10)

d[P]i

dt= k2i[ES]i (11)

with i = 1, . . . , n and initial conditions at t = 0[S]i, [E], [ES]i, [P]i

=

[S0]i, [E0],0,0

. (12)

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Figure 2. Progress curves for an enzymesubstratecompetitor reaction of the form (1) with n = 2.In this reaction a weak competitor (12 1) is assumed. The infinite time range has been reducedto the interval (0, 1) by means of the exponential time scale = 1 1/ ln(t + exp(1)). The reactantconcentrations have been nondimensionally scaled as follows: free substrate si = [S]i/[S0]i; free enzymee = [E]/[E0]; complex ci = [ES]i/[E0] and product pi = [P]i/[S0]i. The subscripts 1 and 2 denote,

respectively, the substrate and weak competitor.

Since the enzyme E is a catalyst, its total concentration (free plus combined)

must be a constant. This conservation law is readily expressed by adding equations (9)

and (10):

d[E]

dt+

ni=1

d[ES]i

dt= 0 [E](t)+

ni=1

[ES]i(t) = [E0]. (13)

Also, at any time the sum of the concentrations of the ith free substrate [S]i, its com-plex [ES]i and product [P]i must be equal to the initial substrate concentration [S0]i;

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Figure 3. Time derivatives for an enzymesubstratecompetitor reaction of the form (1) with n = 2. Inthis reaction a weak competitor (1 1) is assumed. The infinite time range has been reduced to theinterval (0, 1) by means of the exponential time scale = 1 1/ ln(t + exp(1)). The time derivativeshave been scaled to their absolute maximum value: si = d[S]i/dt; e = d[E]/dt; ci = d[ES]i/dt and

pi = d[P]i/dt. The subscripts 1 and 2 denote, respectively, the substrate and weak competitor.

that is, by adding equations (8), (10) and (11) it may be shown that

d[S]i

dt+

d[ES]i

dt+

d[P]i

dt= 0

[S]i(t)+ [ES]i(t)+ [P]i(t) = [S0]i. (14)

Thus the conservation laws reduce the system of differential equations (8)(11) to two

equations for Si and ESi,

d[S]i

dt=ki

[E0]

nj=1

[ES]j

[S]i + ki[ES]i, (15)

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d[ES]i

dt= ki

[E0]

n

j=1[ES]j

[S]i (ki + k2i)[ES]i. (16)

This nonlinear system is complicated to solve, but further simplifications can be intro-

duced if the quasi-steady-state approximation (QSSA) is obeyed.

As previously discussed [5,23,2934,40], the QSSA implies that after initial tran-

sients, t > tCi say, the complex concentrations remain approximately constant. Thatis, in the slow-time regime it can be assumed that

d[ES]i

dt 0, (17)

resulting in the expression

[ES]i =[E0][S

]i

1+j[S]j , (18)where [S]i [S]i/Ki is the reduced concentration with the MM constant being definedas

Ki ki + k2iki

. (19)

Substituting (18) in (11) and simplifying the conservation law (14) with the assump-

tion (17) yields the following differential equation:

d[S]i

dt=

i[S]i1+

j[S

]j, (20)

where

i vmaxi

Ki=

k2i[E0]

Ki(21)

is the first-order rate constant.

The QSSA also implies that during the transient time, t < tCi , there is not asignificant fraction of the substrate bound to the enzyme, that is,

[S]i [S0]i, (22)leading to the important conclusion that (20) is valid in both the fast- and slow-time

regimes. Therefore, solutions valid during the total span of the reaction (0 < t < )can be obtained from the two equations

d[S]i

dt=

i[S]i1+

j[S

]j, (23)

[ES]i=[E0][S

]i

1+

j[S]j

1 exp(it)

, (24)

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where i is the constant

i=

kiKi1+n

j=1[S0]j. (25)Equation (23) can be decoupled by introducing the competition matrix [26,30]

ij ji

(26)

which, for each pair of substrates, provides a measure of their relative rates. We now

obtain the following differential equation:

d[S]j

d[S]i= ij

[S]j

[S]i, (27)

which is easily integrable to result in the useful working relation

[S]j(t)

[S0]j=

[S]i(t)

[S0]i

ij. (28)

Moreover, noting that the competition matrix obeys the following constraints:

ii= 1, (29)

ji =1

ij, (30)

ji =ii

ij, (31)

the reaction system is then completely specified by the row vector

i = (i1, i2, . . . , in). (32)

Equation (28) allows us to conclude that if the competition vector (32) is known for

the ith substrate, then the complete reaction system (i = 1, . . . , n) is characterised bysolely measuring the concentration decay of the ith substrate. The time behaviour ofthe latter can be obtained from the solution of the uncoupled equation

d[S]i

dt=

i[S]i1+

j[S

0]j([S

]i/[S0]i)ij

(33)

with the initial condition ([S]i) = ([S0]i) at t = 0. Conversely, if the time decaysof all the substrates are measured, then the competition matrix can be in principle

reconstructed.

Following [33,34] and generalising the results in [29,30], two sets of time scales

are considered for the reaction system: a set related to the duration of the initial

transients, tCi , and a second containing the times taken for significant changes in

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the Si concentrations, tSi , that provides a measure of the quasi-steady-state period.Using (24), the individual fast-time scales are given by tCi =

1i , namely,

tCi =1

kiKi(1+

j [S0]j)

. (34)

The slow-time scales are determined by dividing the total change in substrate concen-

tration [S0]i by the maximum rate of substrate change |d[S]i/dt|max,

tSi =1+

j[S

0]j

i; (35)

hence, the competition matrix can also be written in terms of the ratio of the slow-time

scales,

ij =tSitSj

. (36)

We are now in a position to establish the general conditions of the QSSA for

the reaction scheme under consideration. Extending the prescription in [33,34] for the

basic enzyme reaction (n = 1) to the multiple substrate case (i = 1, . . . , n), we requirethat

[S]i

[S0]i

tCi

[S0]i

d[S]i

dt

max

1, (37)

which implies that the QSSA is valid if the following condition is obeyed:

maxi=1,...,n

[E0]

Ki(1+

j[S0]j)

1. (38)

4. Solutions

In order to derive closed form solutions for the alternative substrates, we start by

integrating (33) to obtain the relation

it =

nj=1

[S0

]j

ij1 [S]i[S0]i

ij ln [S]i[S0]i. (39)In the case of the single substrate reaction, the usefulness of this progress-curve

equation to estimate reaction constants from experimental data has been recently dis-

cussed [22,29].

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4.1. Even competition (i 1)

In a similar fashion to [30], (39) is rearranged after substituting ij = 1 to obtainthe solution for the time evolution of the reduced concentration of the ith substrate interms of the omega function W,

Si(t) =

[S0]ij[S

0]j

W

j

S0j

exp

it+

j

S0j

. (40)

This relation shows that in even competition the substrate reduced concentrations at

any time are essentially determined by their initial fractional reduced concentrations

and first-order rate constants, thus keeping the proportion

S

1:

S

2:

S

3: :

S

n=

S01

:

S02

:

S03

: :

S0n(41)

throughout the reaction.

4.2. Weak competition (i 1)

For the case where the competition vector i 1, a Taylor expansion showsthat, to first order in ij ,

[S]i

[S0]i

ij= 1+ ij ln

[S]i

[S0]i

+O(ij)

2. (42)

Equation (39) then reduces to

it =

S0i S

i

1+j=i

S0j

ln

[S]i

[S0]i

, (43)

which leads to the following expression for the time evolution of the concentration of

the ith substrate:Si(t) =

1+

j=i

S0j

W

[S0]i

1+

j=i[S0]j

exp

it+ [S0]i1+

j=i[S

0]j

. (44)

Making use of relation (28), the solution for the jth substrate (j = i) is given by

S]j(t) =

S0j

[S]i(t)[S0]i

ij. (45)

Furthermore, (44) can be rewritten asSi(t) = W

S0i

exp it+ S0i, (46)

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where [S]i [S]i/

Ki is the apparent reduced concentration and

i vmaxi /

Ki the

apparent first-order rate constant normalised to the apparent MM constant

Ki = Ki1+j=i

S0j. (47)The ith MM constant displays an increase due to the statistical factor that accountsfor the distribution of enzyme between the E and ESj constituents [32]. Since expres-

sion (46) has a similar form to that for the single substrate reaction (see equation (2)),

the fast competitor substrate behaves as apparently alone, and the competition is

adequately taken into account by means of the apparent MM constant.

4.3. Mixed competition

Rather than considering the specific case of strong competition, which as pre-

viously discussed is trivial, we consider the more general case of mixed competitionwith l evenly competitive alternative substrates substrates, ij 1 for j = 1, . . . , l, inthe presence of (n l) weak competitor substrates, ij 1 for j = (l + 1), . . . , n.The concentration decay of the i = 1, . . . , l competitor substrates is now expressed as

Si(t)=

[S0]ilj=1[S

0]j

1+

nj=l+1

S0j

W l

j=1[S0]j

1+n

j=l+1[S0]j

exp

it+lj=1[S0]j1+

nj=l+1[S

0]j

(48)

while the concentrations of the weak competitor substrates, j = (l + 1), . . . , n, can

again be derived from relation (28):Sj

(t) =

S0j

[S]i(t)

[S0]i

ij. (49)

The effect of the (n l) weak competitor substrates can again be incorporated in theapparent MM constant

Ki = Ki

1+

nj=l+1

S0j

(50)

to provide simplified expressions for the apparent reduced concentrations of the nsubstrates:

Si(t)=

[S0 ]ilj=1[S

0 ]j

W l

j=1

S0j

exp

it+ l

j=1

S0j

(i l), (51)

Sj

(t)=

S0j

[S]i(t)

[S0 ]i

ij(i > l). (52)

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5. Discussion

We have developed a general and compact formalism that fully describes the

reaction of multiple alternative substrates. The analytic solutions that have been de-

rived for each case of competition are certainly a considerable advance from previous

work that mainly relied on involved graphical methods or numerically integrated equa-

tions. From a qualitative point of view, such solutions bring out in a clear fashion the

collective kinetic behaviour of a set of multiple substrates competing for an enzyme.

Furthermore, as recently illustrated [22], they can also be exploited to implement non-

linear least-squares fitting procedures that can result in estimates of reaction constants

of quantitative reliability. It is worth emphasising that since the proposed curve fitting

is aimed at concentration decays rather than velocities, experimental methodologies

should be greatly simplified. Since the present work is part of a systematic revision

of enzyme reaction networks, we are confident that the formalism can be extended to

tackle multimode inhibition enzyme kinetics that will be reported elsewhere.

Appendix

The omega function, W(x), is defined as the function satisfying the followingtranscendental equation [13,42]:

W(x)exp

W(x)= x for x exp(1). (53)

A plot of W(x) (see figure 4) shows that for exp(1) x < 0 the function hastwo possible real values. From these dual values, two real branches of W(x) canbe defined: the upper branch W(x)

1 and the lower branch W(x)

1. For

convenience, we only consider here the upper branch.

Figure 4. Representation of the W(x) function, showing the upper branch (solid curve) and the lowerbranch (dotted curve).

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By rewriting expression (53) as W(x) = x/ exp(W(x)), the omega function Wcan be expressed as a continued fraction for |W(x)| < 1,

W(x) = xexp(W(x))

= xexp

x

exp(W(x))

= = xexp

x

expx

expx

. . .

(54)

or, it can be rewritten as W(x) = ln(x/W(x)) leading to the alternative continuedfraction for |W(x)| > 1,

W(x) = lnx

ln(W(x))= ln

x

lnx

ln(W(x))

= = ln xln

x

lnx

lnx

. . .

. (55)

The exponential continued fraction (54) can be used to approximate W(x) aroundx = 0; however, a polynomial continued fraction approximation is more effective [2,3].A series expansion of equation (53) at x = exp(1) yields

W(x) = 1+v

1

3 11

72

v

v

43

540 769

17280

v

v2 +O

v3

, (56)

where v = 2+ 2 exp(1)x. Converting this series into a continued fraction yields

W(x) = 1+

v

1+

v

3+m

, (57)

where

m =a

v

b+

v. (58)

Adjusting a and b to minimise the error in the interval exp(1) x 2 gives

a=4 32+ b(22 3)

2 2 , (59)

b=28769

6237 v +17035

155491/4

. (60)

For x > 2, the following truncated form of the logarithmic continued fraction (55)also provides a satisfactory approximation:

W(x) = lnx

lnx

lnn(x)

, (61)

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where

n = expc

d+ ln(x) (62)with

c =42887

38139, d =

9737

23046. (63)

Acknowledgements

Part of this work was carried out during a visit of S. Schnell to the Physics

Center of the Instituto Venezolano de Investigaciones Cientficas (IVIC) in March

1999 and January 2000. The present work has been partially funded by CONICIT

under contract G-97000667. SS is the recipient of the Jose Gregorio Hernandez Award

(Academia Nacional de Medicina, Venezuela, and Pembroke College, Oxford, UK), ofthe ORS Award (Committee of Vice-Chancellors and Principals of the Universities the

United Kingdom), of a Lord Miles Senior Scholarship in Science (Pembroke College)

and participant of the Programa de Cofinanciamiento Institucional IVIC-CONICIT

(Venezuela). We are also grateful to Professor P.K. Maini for a critical revision of the

manuscript.

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